One trending application of LLM (large language model) is to use it for content moderation in online platforms. Most current studies on this application have focused on the metric of accuracy -- the extent to which LLMs make correct decisions about content. This article argues that accuracy is insufficient and misleading because it fails to grasp the distinction between easy cases and hard cases, as well as the inevitable trade-offs in achieving higher accuracy. Closer examination reveals that content moderation is a constitutive part of platform governance, the key of which is to gain and enhance legitimacy. Instead of making moderation decisions correct, the chief goal of LLMs is to make them legitimate. In this regard, this article proposes a paradigm shift from the single benchmark of accuracy towards a legitimacy-based framework for evaluating the performance of LLM moderators. The framework suggests that for easy cases, the key is to ensure accuracy, speed, and transparency, while for hard cases, what matters is reasoned justification and user participation. Examined under this framework, LLMs' real potential in moderation is not accuracy improvement. Rather, LLMs can better contribute in four other aspects: to conduct screening of hard cases from easy cases, to provide quality explanations for moderation decisions, to assist human reviewers in getting more contextual information, and to facilitate user participation in a more interactive way. To realize these contributions, this article proposes a workflow for incorporating LLMs into the content moderation system. Using normative theories from law and social sciences to critically assess the new technological application, this article seeks to redefine LLMs' role in content moderation and redirect relevant research in this field.